Controllability of semi-linear piecewise fractional delayed dynamical systems

This paper presents an exploration of linear piecewise fractional delayed systems, aiming to derive an explicit solution with the help of mathematical induction. We establish the existence and uniqueness of global solutions in the semilinear context through the application of the Banach fixed-point theorem, while also addressing the stability of these semilinear systems. Furthermore, we outline both sufficient and necessary conditions for the controllability of linear control systems utilising the Gramian matrix, and we identify sufficient conditions for the controllability of semilinear control systems through the Schaefer fixed-point theorem. Numerical examples are presented to support our findings.